Unveiling phase diagram of the lightly doped high-Tc cuprate superconductors with disorder removed

The currently established electronic phase diagram of cuprates is based on a study of single- and double-layered compounds. These CuO2 planes, however, are directly contacted with dopant layers, thus inevitably disordered with an inhomogeneous electronic state. Here, we solve this issue by investigating a 6-layered Ba2Ca5Cu6O12(F,O)2 with inner CuO2 layers, which are clean with the extremely low disorder, by angle-resolved photoemission spectroscopy (ARPES) and quantum oscillation measurements. We find a tiny Fermi pocket with a doping level less than 1% to exhibit well-defined quasiparticle peaks which surprisingly lack the polaronic feature. This provides the first evidence that the slightest amount of carriers is enough to turn a Mott insulating state into a metallic state with long-lived quasiparticles. By tuning hole carriers, we also find an unexpected phase transition from the superconducting to metallic states at 4%. Our results are distinct from the nodal liquid state with polaronic features proposed as an anomaly of the heavily underdoped cuprates.

. As the hole doping decreases, the Fermi arc shrinks and eventually becomes point nodes (or nodal liquid state) at the edge of the Mott insulating phase [11][12][13][14][15] . In this state, the quasiparticle peak is tiny and accompanied by polaron-like broad spectra. In addition, this peak disappears with leaving off the node since very broad spectra severely damped by the pseudogap prevail everywhere in momentum space. At doping levels further less, a gap is opened even in the (0,0)-(π, π) direction 13,14,16 , turning the nodal liquid state into a full gap state consisting only of polaronic broad spectra all around the Brillouin zone (BZ).
Notably, the above phase diagram is based on the data of singleand double-layered cuprates; it is different from the phase diagrams with a significant overlap of the AF and SC phases in multilayer systems with three or more CuO 2 planes per unit cell 17,18 . In the single-and double-layered cuprates, the CuO 2 plane is affected by the random potential induced by the adjacent dopant layers, leading to an inhomogeneous electronic state as revealed by scanning tunneling microscopy (STM) 19,20 . It is, therefore, possible that the phase diagram is relevant only for disordered CuO 2 planes, especially in the lightlydoped region sensitive to disorder. This circumstance may have hindered a fair comparison of the data with the theory describing the doped Mott state, which usually supposes an ideal CuO 2 plane without disorder 21,22 . It could, however, be solved by focusing as a research target on the inner planes of the multilayer cuprates, which are protected by the outer CuO 2 planes screening the disorder effect from the dopant layers. According to nuclear magnetic resonance (NMR) studies 17,18 , the carrier doping in the inner planes is much more homogeneous than that in the CuO 2 planes of other compounds, including the single-layered HgBa 2 CuO 4+δ (Hg1201) and doublelayered YBa 2 Cu 3 O 6.5 (Y123) thought as to be clean systems. With this advantage, the small Fermi Pocket, which had been elusive while predicted in the doped Mott state, was recently observed in the inner plane of a 5-layer compound 23 . The multilayer cuprates, therefore, provide an excellent platform to unveil the genuine electronic properties of the lightly-doped region, which is key to elucidating the pairing mechanism in cuprates. Above all, since the highest achievable T c among the existing substances is obtained in one of the multilayer cuprates (the trilayer HgBa 2 Ca 2 Cu 3 O 8+δ [24][25][26] ), the current subject is crucial for the development of condensed matter physics.
In this article, we have selected the six-layer Ba 2 Ca 5 Cu 6 O 12 (F,O) 2 (T c = 69 K; Supplementary Fig. S1) for a study, where the effective carrier doping of the inner planes should be very low. The electronic properties of the clean CuO 2 planes are revealed over a wide range of hole doping which is controlled by varying the number of inner planes and the in situ potassium deposition on the sample surface. The spectra with well-defined quasiparticle peaks lacking the polaronic features are detected all over the closed Fermi surface (or a tiny Fermi pocket), even at the doping level extremely close to the half-filling; it is distinct from the nodal liquid state and the polaronic state established in the heavily underdoped cuprates with the inevitable disorder. Furthermore, we find that the superconducting pairing occurs at~4% doping, almost the same critical doping as in single-layered cuprates with CuO 2 planes severely disordered. This doping level (~4%), therefore, is not the consequence of an increase by the disorder but should be the critical amount of carriers essential for the pair formation even in the ideally clean CuO 2 plane. Figure 1a plots the spectral intensities close to the Fermi level (E F ) measured by laser-ARPES at the lowest temperature (T = 5 K). We found three sheets of FSs: One exhibits an arc-like structure typical for the underdoped cuprates, and the other two show small pockets around (π/2, π/2) corresponding to the doped Mott states with the AF order 27 . To further validate the ARPES results, we also observed the de Haas-van Alphen (dHvA) effect by the torque measurement, a bulksensitive probe. We detected quantum oscillations (Fig. 1c) consisting of mainly two frequencies (arrows in Fig. 1f), corresponding to the FS areas covering 1.2 and 4.8% of the Brillouin zone. These values almost perfectly agree with the ARPES results (1.0 and 4.3%). Since carriers are doped from the dopant layers (hatched by orange in Fig. 1f), the doping amount should become less toward the inner planes. It is, thus, expected that the small Fermi pocket, large Fermi pocket, and Fermi arc are each formed by the innermost layer (IP 0 ), second-inner plane (IP 1 ), and outer plane (OP), respectively, as noted in Fig. 1a, f. The validity of this one-to-one correspondence between FSs and CuO 2 layers can be confirmed by comparing these results with those of the five-layer compound 23 , as demonstrated below.

Results
In Fig. 1b, we overlay the FSs of the fiveand six-layer compounds determined from the laser-ARPES data. Here, note that the samples we observed have similar T c values (T c = 65 K and 69 K for fiveand sixlayer compounds, respectively), and thus these two should have similar doping levels. We found that the small Fermi pocket gets much smaller than that of the five-layer compound, while the other FSs (large Fermi pocket and Fermi arc) remain almost the same. We also obtained the results supporting this conclusion by synchrotron-ARPES (Supplementary Fig. S2) with higher photon energies more generally used in the cuprate research. As summarized in Fig. 1e, the area of the small Fermi pocket labeled as IP 0 is changed by adding one more CuO 2 plane in the unit cell from five to six, and importantly, the area of the 6-layer compound gets half that of the 5-layer compound (arrows in Fig. 1e). This indicates that the carriers in IP 0 are simply split into two for IP 0 s doubled in the 6-layer compound (arrows in Fig. 1f) without affecting other planes (IP 1 and OP); this means that the wave function of IP 0 is independent of those of IP 1 and OP. It is further justified by our experimental result that the superconducting gap is observed on the Fermi pocket of IP 1 but not of IP 0 ( Supplementary Fig. S4). The mixing of layers should produce superconducting gaps of similar magnitudes, so our data against it indicate that the two pockets derive each from different layers (IP 0 or IP 1 ) that are essentially electronically decoupled.
We perform a model calculation based on our ARPES results and demonstrate that the mixing of layers is, indeed, negligible (Supplementary Fig. S3). The hybridization among layers is prevented by a potential difference induced by the carrier distribution along the caxis. The inclusion of an interlayer hopping parameter (V) in the calculations doubly splits the small Fermi pocket, similar to the bilayer splitting observed in double-layer cuprates 28 . This is attributed to double IP 0 s adjacent to each other and sensitive to the V parameter. In contrast, splitting does not appear in the large Fermi pocket even for a relatively large value of V since double IP 1 s are structurally separated. Notably, splitting of the small Fermi pocket (IP 0 ) is not experimentally observed both in ARPES and dHvA. We also note that the peak of the fast Fourier transformation (FFT) spectrum of the dHvA effect for the small Fermi pocket is relatively sharp in width (at least, sharper than the spectra of Y123 29 and Hg1201 30 ), and it is almost the same as that for the large Fermi pocket. It indicates that there is not even the slightest splitting in the small Fermi pocket, so the interlayer hopping should be negligibly small in our samples. This is a remarkable feature of a low doping state, and compatible with the observation in Bi2212 that the bilayer splitting energy gets smaller with decreasing carrier concentration 31 . Calculations with such a small V estimate the mixture of wave functions from different CuO 2 planes to be negligible (less than 3%; Supplementary Fig. S3d). This leads us to conclude that the three FSs we observed are independently formed by three different CuO 2 planes (IP 0 , IP 1 , and OP). We emphasize that this is a new aspect of multilayered cuprates that was clarified only through the measurement of the 6-layer compounds with double IP 0 s.
The above argument allows us to estimate the carrier concentration of each CuO 2 plane directly from the area of each FS. Our data indicate that the innermost CuO 2 plane (IP 0 ) is doped by only 1% of hole carriers, which is extremely close to the half-filling. In Fig. 2b, c, we plot the energy distribution curves (EDCs) around the small Fermi pocket for IP 0 (orange circles in Fig. 2a) and those symmetrized about E F to eliminate the Fermi cut-off effect, respectively. Surprisingly, we find very sharp peaks in the spectra even for a state with such low carrier density. Here, note that the superconducting gap is absent most likely because the state of 1% doping is situated outside the superconducting dome in the phase diagram. The spectral peak width is estimated to be about 7.1 meV (arrows in Fig. 2c), which is comparable to or even smaller than that (blue curve in Fig. 2c) of the optimally doped Bi 2 Sr 2 CaCu 2 O 8+δ (Bi2212) 32 , the cuprate material most well-studied by ARPES. This means that quasiparticles as long-lived as those in the optimally doped state can develop with a tiny amount of carrier doping in an ideally clean CuO 2 plane.
Notably, the peak width is constant all around the Fermi pocket including hot spots at which the FS and the antiferromagnetic zone boundary (AFZB) cross 33,34 . This is very different from the anisotropic nature widely acknowledged for the underdoped cuprates 3 . We also emphasize that the electronic state unveiled here is distinct from the following features illustrated for the single-and double-layered cuprates: the nodal liquid state, where only the nodal direction is metallic and the other k F points are dominated by broad spectra with the pseudogap [11][12][13][14] , and the polaronic state, where quasiparticle peaks are tiny and largely buried by a hump-shaped incoherent part. In stark contrast, our data for inner planes exhibit surprisingly simple metallic features, forming a closed Fermi pocket (instead of the nodal liquid state) with well-defined quasiparticles (instead of the polaronic state). Here, note that the quasiparticle in IP 0 is not a product of the superconducting proximity effect from the outer planes. This is evidenced by the fact that quantum oscillations (signals of well-defined quasiparticles) have been observed under conditions that the superconductivity is completely suppressed. ARPES also confirmed that the    h The EDC at k F for IP 0 , indicating a well-defined quasiparticle peak accompanied by the spectral tail with relatively low intensities lacking polaronic features.
quasiparticle peak persists even above T c although its width gets broadened due to the thermal broadening effect ( Supplementary  Fig. S10). The bands of Fermi arc and pockets are mutually close in momentum space, so their ARPES signals could interfere at high binding energies. To examine single-particle spectra of IP 0 , we conducted band-selective measurements by utilizing the matrix element effect. In Fig. 2e, we map the FS by synchrotron-ARPES not only of the 1st BZ but also up to the 2nd BZ. The data in the 2nd BZ are enhanced in intensity, and moreover, separate the bands of Fermi arc (OP) and pockets (IP 0 and IP 1 ) in the upper and lower half of the 2nd BZ, respectively. This separation is further confirmed in Fig. 2f, g by plotting the ARPES dispersions each across nodal k F points of OP and IP 0 (dashed lines in Fig. 2e). Only the latter exhibits the folded bands about AFZB, which are also revealed by MDCs at E F (the upper panels of Fig. 2f, g). We extract the EDC for IP 0 at k F in Fig. 2h, and find a sharp peak accompanied by a tail with relatively low intensities up to the energy scale of the bandwidth. This validates that the Fermi pocket possesses well-defined quasiparticles lacking polaronic features, even though the doping level is extremely small.
In order to explore a wider doping range, we have performed the in situ potassium deposition on the samples. This technique is commonly used in ARPES 35 including a study of cuprates [36][37][38] . Figure 3a-c displays the FS mapping before and after the potassium deposition for different times (0, 30, and 60 s, respectively). The Fermi pockets get smaller with deposition time. This variation is more clearly demonstrated in Fig. 3f by extracting the momentum distribution curves (MDCs) along the AFZB at E F . The momentum distance between each paired k F s becomes shorter in the two pockets, representing the reduction of hole carriers in both inner planes. In Fig. 3d, e, we plot the large and small Fermi pockets for IP 1 and IP 0 , respectively, at three different deposition times, determined from the ARPES spectra. A systematic shrinkage of the pockets is confirmed, as estimated in Fig. 3g from their areas (or carrier concentrations ps). The p value  Fermi surface mapping zoomed around (π, π) to focus on the two Fermi pockets for three cases: before deposition (a) and after deposition for 30 s (b) and 1 min (c). The spectral intensities were integrated within an energy window of 10 meV around E F . d, e Fermi pockets determined from the peak positions of MDCs for IP 1 (d) and IP 0 (e), respectively. In each panel, the results of three different deposition times (0 s, 30 s, and 60 s) are plotted. f Comparison of MDCs at E F along the antiferromagnetic zone boundary (AFZB). Along this momentum cut, the Fermi velocity is fastest, thus the largest shift of k F with carrier doping is expected. g The deposition time dependence of the area of the Fermi pocket or the hole carrier concentration p. The dotted line remarks that IP 0 reached the doping level as extremely low as 0.7 % (p = 0.007) after the deposition for 60 s. decreases faster in the large pocket (IP 1 ) than in the small pocket (IP 0 ); this is expected since IP 1 lies closer to the surface dopant layer where potassium is deposited, so it should be more efficiently doped. Our experiments could reduce the doping level of the innermost plane (IP 0 ) down to 0.7% (p = 0.007), which is so small as to nearly reach the half-filled Mott state.
Here we examine the doping evolution of the ARPES spectra. Figure 4a plots the EDCs at k F on the AFZB for IP 0 (orange circle in the inset of Fig. 4a) in 1.0, 0.9, and 0.7% of doping levels. In the right panel, the symmetrized EDCs are also plotted. We found that sharp quasiparticle peaks persist down to the lowest carrier concentration of 0.7% (p = 0.007). Although the peak intensity was slightly suppressed due to the sample surface deterioration, the spectral peaks have almost the same width ( Supplementary Fig. S8), indicating that the scattering rate (or lifetime) of quasiparticles is hardly changed going toward the halffilling. The results imply that a single hole doped into the Mott insulator can behave as a long-lived quasiparticle in a clean CuO 2 plane (Fig 4b). It is in stark contrast to the property of the single-and doublelayered cuprates, in which a quasiparticle peak rapidly dies out at doping levels lower than~10% 13,14,16,37 , and even if there is some spectral weight at E F , it is rather broad and disappears immediately with going away of the nodal direction 12,39 . IP 1 also realizes a very clean electronic system as confirmed by its low Dingle temperature (T D , proportional to the scattering rate)  Fig. 4 | Doping evolution of ARPES spectra and phase diagram of the lightlydoped cuprates with disorder removed. a Doping evolution of the energy distribution curves (EDCs) and those symmetrized about E F , measured at k F on the antiferromagnetic zone boundary (AFZB) for the small Fermi pocket (orange circle at the inset). The spectra plotted are for three different doping levels (1, 0.9, and 0.7%) controlled by potassium deposition. b Schematic of electronic state in real space for the 1% doped CuO 2 plane, illustrated based on our data suggesting that a single hole can behave as a long-lived quasiparticle in the antiferromagnetic background. c Schematic density of states for the 1% doped CuO 2 plane corresponding to b. d-f Doping evolution of EDCs and those symmetrized about E F for three different k F s (purple circles in the inset of each panel) of the large Fermi pocket. The spectra plotted are for three different doping levels (4.3, 4.0, and 3.6%) controlled by potassium deposition. The arrows point to the spectral peaks, clarifying the gap opening or closing for each spectrum. g The phase diagram in lightlydoped region summarized from our data of clean inner CuO 2 planes in 6-layered cuprates. AF, QP, and SC abbreviate "antiferromagnetic'', "quasiparticle'', and "superconductivity'', respectively. obtained by the dHvA quantum oscillations: T D of IP 1 is estimated to be 12.3 K (Supplementary Fig. S5), which is between those of Y123 (T D = 6.2 K) 29,40 and Hg1201 (T D = 18 K) 30,40 . Note that the carrier concentration of IP 1 directly estimated from the Fermi pocket area is small, only to be 4.3%, which is less than half those (~10%) of Y123 and Hg1201 samples used for the quantum oscillation measurements. Hence, the T D of IP 1 is rather small, considering that it is obtained in such a low carrier concentration with a poor screening effect.
The spectra of IP 1 before the potassium deposition exhibit the superconducting gap consistent with the d-wave symmetry, being the largest at the tip of the Fermi pocket (see Supplemental Fig. S4 for more details). This data indicates the coexistence of superconductivity and AF magnetic order, similar to the observations in the five-layer compound by NMR 17,18 and ARPES 23 . Figure 4f plots the spectra at the tip of the Fermi pocket for three different doping levels (4.3, 4.0, and 3.6%) controlled by the potassium deposition. The superconducting gap observed at the original doping level of 4.3% closes at 4.0% (This is further justified in Supplementary Note 9). This indicates that the electronic state in IP 1 has got out of the superconducting dome and entered the metallic phase, i.e., the same as that of IP 0 , by decreasing the hole doping. Also, the relatively small superconducting gap observed for the pristine surface (magnitude of Δ tip = 5 meV at the tip of the Fermi pocket and the order parameter Δ 0 = 11 meV determined by extrapolating it to the antinode) is attributed to the carrier doping level (4.3%) located almost at the edge of the superconducting dome. The superconducting to metal transition we observed occurs by closing a gap along the entire Fermi surface (or pocket) while maintaining well-defined quasiparticles all over it. This is very different from the phase transition for the underdoped cuprates with disordered CuO 2 planes, where the pseudogap accompanied by the damped broad spectra plays a significant role. Rather, our observation is quite similar to the doping-induced phase transition across the superconducting dome on the overdoped side with no indication of the pseudogap, although the Fermi surface size is apparently different, corresponding to p, not to 1+p.
In Fig. 4g, we illustrate a phase diagram of lightly-doped cuprates with extremely clean CuO 2 planes unveiled via the direct observation of the electronic structure by ARPES. The Mott insulating state is realized only when the CuO 2 plane is non-doped, and is strictly half-filled; only the slightest amount of hole doping changes it to a metallic state forming a Fermi pocket with well-defined quasiparticles on the top of the lower Hubbard band (or charge transfer band). We find that the effective mass in the innermost layer (IP 0 ) and the second-inner layer (IP 1 ) with different carrier concentrations (~1 and~4%, respectively) are almost the same (~0.6 m 0 ) by quantum oscillation measurements ( Supplementary Fig. S7). This indicates a lack of a pronounced band narrowing when varying p toward the half-filled Mott state. The transition to a Mott insulator most likely occurs by completely removing hole carriers from the lower Hubbard band until the perfect half-filling, rather than by controlling the bandwidth. At 4%, the metal-tosuperconductor transition occurs by opening the superconducting gap, and the system enters the phase where the AF order and superconductivity coexist. Intriguingly, this critical doping level is almost the same as that of some single-and double-layer compounds, such as La 2−x Sr x CuO 4 which is known to be severely disordered according to the NMR studies [41][42][43][44] . This indicates that 4% is the intrinsic critical doping level necessary to form the superconducting pairs even in the CuO 2 planes with the clean-limit condition. This scenario differs from theories that suggest that superconductivity occurs simultaneously with the appearance of metallicities by carrier doping to the ideal Mott insulator without disorder 21,22 . Our results will provide crucial insight into understanding the intrinsic relationship between the Mott physics and the pairing mechanism in the lightly-doped CuO 2 planes, which has not been accessible for the single-and double-layered compounds mainly studied in the long history of cuprate research.

Samples
Single crystals of underdoped Ba 2 Ca 5 Cu 6 O 12 (F,O) 2 (see crystal structure in Fig. 1f) with T c = 69 K were grown at between 1100 and 1200 ∘ C under a pressure of 4.5 GPa without an intentional flux. The starting composition for the crystal synthesis is Ba 2 Ca 3 Cu 4 O 7.9 F 2.1 , which is known to be almost the same in single crystals 45 . We have conducted X-ray diffraction measurements along the c-axis for all the sample pieces and confirmed that they are single crystals, not mixtures of crystal domains with different numbers of CuO 2 layers per unit cell. Magnetic susceptibilities for these crystals (Fig. S1) show a sharp superconducting transition with~4 K in width, indicative of high quality in our samples; the signal-to-noise ratio is not so high owing to the small volume in our crystals (~300 × 300 × 50 μm in crystal size). Laue image of the single crystal (Fig. S1b) shows a four-fold rotational symmetry with no indication of structural modulations.

ARPES measurements
Laser-based ARPES data were accumulated using a laboratory-based system consisting of a Scienta R4000 electron analyzer and a 6.994 eV laser (the sixth harmonic of Nd:YVO 4 quasi-continuous wave). The data presented are measured at 5 K. The overall energy resolution in the ARPES experiment was set to 1.4 meV. Synchrotron-based ARPES measurements were performed at a highresolution branch (HR-ARPES) of the beamline I05 in the Diamond Light Source, equipped with a ScientaOmicron R4000 analyzer. The data presented are measured at the photon energy of 55 eV and at the temperature of 10 K. The overall energy resolution was set tõ 10 meV in our experiments. In both the laser-and synchrotron-ARPES measurements, a typical cleavage method was used to get a clean surface of the samples: a top post glued on the crystal is hit in situ to obtain a flat surface suitable for the ARPES measurements. The cleavage plane has been confirmed by STM to be along the F/O dopant layers 45 .

Quantum oscillation measurements
Torque magnetometry experiments were performed with a commercial piezoresistive cantilever (SEIKO PRC-120) 46 in pulsed magnetic fields up to 60 T (36 ms pulse duration). The cantilever directly detects the magnetic torque (τ) as the result of the anisotropic magnetization of the sample, τ = M × H, and the magnetic quantum oscillation known as the de Haas-van Alphen (dHvA) oscillation was observed. Figure 1c in the main paper shows the data after subtracting background, which was obtained by fitting a quadratic function to each curve of the raw data in the range of magnetic field between 26 and 60 T.

Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.